Skip to main content
SpringerLink
Log in

On the Discriminant Scheme of Homogeneous Polynomials

  • Published:
Mathematics in Computer Science Aims and scope Submit manuscript

Abstract

In this paper, the discriminant scheme of homogeneous polynomials is studied in two particular cases: the case of a single homogeneous polynomial and the case of a collection of n − 1 homogeneous polynomials in \({n\geqslant 2}\) variables. In both situations, a normalized discriminant polynomial is defined over an arbitrary commutative ring of coefficients by means of the resultant theory. An extensive formalism for this discriminant is then developed, including many new properties and computational rules. Finally, it is shown that this discriminant polynomial is faithful to the geometry: it is a defining equation of the discriminant scheme over a general coefficient ring k, typically a domain, if \({2\neq 0}\) in k. The case where 2 = 0 in k is also analyzed in detail.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Apéry, F., Jouanolou, J.-P.: Élimination: le cas d’une variable. Hermann, Collection Méthodes (2006)

  2. Benoist O.: Degrés d’homogénéité de l’ensemble des intersections complètes singulières. Ann. Inst. Fourier (Grenoble) 62(3), 1189–1214 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. Busé L., Mourrain B.: Explicit factors of some iterated resultants and discriminants. Math. Comp. 78(265), 345–386 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bruns, W., Vetter, U.: Determinantal rings. In: Lecture Notes in Mathematics, vol. 1327. Springer, Berlin (1988)

  5. Demazure, M.: Résultant, discriminant. Unpublished Bourbaki manuscript, July (1969)

  6. Demazure M.: Résultant, discriminant. Enseign. Math. (2) 58(3–4), 333–373 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. Eisenbud, D., Harris, J.: The geometry of schemes. In: Graduate Texts in Mathematics, vol. 197. Springer, New York (2000)

  8. Esterov A.: Newton polyhedra of discriminants of projections. Discrete Comput. Geom. 44(1), 96–148 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  9. Gel’fand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, resultants, and multidimensional determinants. In: Mathematics: Theory & Applications. Birkhäuser Boston Inc., Boston (1994)

  10. Hartshorne, R: Algebraic geometry. In: Graduate Texts in Mathematics vol. 52. Springer, New York (1977)

  11. Henrici, O: On certain formulæ concerning the theory of discriminants. In: Proceedings of the London Mathematical Society, pp. 104–116, Nov. 12th (1868)

  12. Jouanolou J.-P.: Le formalisme du résultant. Adv. Math. 90(2), 117–263 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  13. Jouanolou, J.-P.: Résultant et résidu de grothendieck. Prépublication IRMA Strasbourg (1992)

  14. König J.: Einleitung in die allgemeine Theorie der algebraischen Groessen. Teubner, Leipzig (1903)

    MATH  Google Scholar 

  15. Kronecker L.: Grundzuege einer arithmetischen theorie der algebraischen groessen. J. Math. 92, 1–122 (1882)

    MATH  Google Scholar 

  16. Krull W.: Funktionaldeterminanten und Diskriminanten bei Polynomen in mehreren Unbestimmten. Monatsh. Math. Phys. 48, 353–368 (1939)

    Article  MathSciNet  Google Scholar 

  17. Krull W.: Funktionaldeterminanten und Diskriminanten bei Polynomen in mehreren Unbestimmten. II. Monatsh. Math. Phys. 50, 234–256 (1942)

    Article  MathSciNet  Google Scholar 

  18. Mertens F.: Ueber die bestimmenden eigenschaften der resultante von n formen mit n veraenderlichen. Sitzungsberichte der Wiener Akademie 93, 527–566 (1886)

    MATH  Google Scholar 

  19. Mertens, F.: Zur theorie der elimination. Sitzungsberichte der Wiener Akademie 108, 1173–1228 and 1344–1386 (1892)

  20. Ould Mohamdi, El Khalil: Élimination Réduite. PhD thesis, University of Strasbourg (1988)

  21. Ostrowski A.: Beweis der Irreduzibilität der Diskriminante einer algebraischen Form von mehreren Variablen. Math. Z. 4(3–4), 314–319 (1919)

    Article  MATH  MathSciNet  Google Scholar 

  22. Saito T.: The discriminant and the determinant of a hypersurface of even dimension. Math. Res. Lett. 19(4), 855–871 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  23. Scheja G., Storch U.: Anisotropic discriminants. Comm. Algebra 36(9), 3543–3558 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  24. Sylvester. Addition à la note sur une extension de la théorie des résultants algébriques. Comptes Rendus de l’Académie des Sciences. LVIII, 1178–1180 (1864)

  25. Sylvester. Sur l’extension de la théorie des résultants algébriques. Comptes Rendus de l’Académie des Sciences. LVIII, 1074–1079 (1864)

  26. Zariski O.: Generalized weight properties of the resultant of n + 1 polynomials in n indeterminates. Trans. Am. Math. Soc. 41(2), 249–265 (1937)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. INRIA Sophia Antipolis-Méditerranée, 2004 routes des Lucioles, B.P. 93, 06902, Sophia Antipolis, France

    Laurent Busé

  2. Université Louis Pasteur, 7 rue René Descartes, 67084, Strasbourg Cedex, France

    Jean-Pierre Jouanolou

Authors
  1. Laurent Busé
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jean-Pierre Jouanolou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Laurent Busé.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Busé, L., Jouanolou, JP. On the Discriminant Scheme of Homogeneous Polynomials. Math.Comput.Sci. 8, 175–234 (2014). https://doi.org/10.1007/s11786-014-0188-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11786-014-0188-7

Keywords

Search

Navigation